
%!TEX program = xelatex
%!TEX TS-program = xelatex
%!TEX encoding = UTF-8 Unicode

\documentclass[10pt]{article} 

\input{wang_preamble.tex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{titling}
\setlength{\droptitle}{-2cm}   % This is your set screw

%%文档的题目、作者与日期
\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{复变函数练习1.3-1.4}
%\date{\vspace{-3ex}}
\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
\date{2024 年 3 月 11 日}
%\date{March 9, 2021}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 01
下述函数哪些是单值函数，哪些是多值函数？
\begin{enumerate}[label={(\arabic*)}]
\item  $w=|z|$. 
\item  $w=\bar{z}$. 
\item  $w=z^2$. 
\item  $w=\frac{z+1}{z-1},\,\, z\neq 1$. 
\item  $z=\sqrt[n]{z}, \,\, n\ge 2$. 
\item  $w=\mathrm{Arg}(z)$. 
\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 02
将复变函数 $w=z^2+2$ 写成代数形式和三角形式。

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 03
写出从复数点集 $E$ 到复数点集 $F$ 的一一映射 $w=f(z)$ 的定义。考察函数 $w=\bar{z}$ 所构成的映射。

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 04
设有函数 $w=z^2$, 它将 $z$ 平面上的下述曲线变成 $w$ 平面上的何种曲线？  
\begin{enumerate}[label={(\arabic*)}]
\item  以原点为圆心，2为半径的在第一象限的圆弧。
\item  倾角为 $\theta=\pi/3$ 的直线。
\item  双曲线 $x^2-y^2=4$. 
\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 05
什么是复变函数 $w=f(z)$ 当 $z\to z_0$ 时的极限？
%记 $z=x+iy$, 写出复变函数 $f(z) = u(x,y) + iv(x,y)$ 在 $z\to z_0$ 时的极限为 $a+ib$ 的充分必要条件。

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 06
证明函数 $f(z) = \frac{z}{\bar{z}}$ 当 $z\to 0$ 时的极限不存在。

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 07
证明函数 $f(z) = \frac{1}{2i} \left( \frac{z}{\bar{z}} - \frac{\bar{z}}{z} \right)$ 在 $z\to 0$ 时的极限不存在。

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 08
什么时候称复变函数 $w=f(z)$ 在 $z_0$ 连续？
什么时候称复变函数 $w=f(z)$ 在点集 $E$ 上连续？


\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 09
设 $\lim\limits_{z\to z_0}f(z)=\eta$, 证明函数 $f(z)$ 在 $z_0$ 的某个去心邻域内是有界的。

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 10
设函数 $f(z)$ 在点 $z_0$ 连续，且 $f(z_0)\neq 0$. 证明 $f(z)$ 在 $z_0$ 的某个邻域内恒不为零。


\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 11
考虑定义在有界闭集 $E=\{z:|z|\le 1\}$ 上的复变函数 $f(z)=\frac{1}{2-z}$. 
\begin{enumerate}[label={(\arabic*)}]
\item  验证 $f(z)$ 在 $E$ 上是连续的。
\item  求 $|f(z)|$ 在 $E$ 上的最大值与最小值。
\item  验证 $f(z)$ 在 $E$ 上是一致连续的。
\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 12
证明函数 
$$f(z)=\left\{
\begin{array}{ll}
\frac{1}{z}, & z\neq 0, \\
\infty, & z=0, \\
0, & z=\infty,
\end{array}\right.
$$  
在扩充复平面上是广义连续的。

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{enumerate}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


